Step 1: Understand what sin^(-1) means. It is the inverse function of sin, also known as arcsin.
Step 2: Identify the angle given in the question, which is 5π/6.
Step 3: Check if 5π/6 is within the range of the arcsin function. The range of arcsin is from -π/2 to π/2.
Step 4: Since 5π/6 is greater than π/2, it is not in the range of arcsin.
Step 5: Find the equivalent angle of 5π/6 that is within the range of arcsin. The sine function is positive in the second quadrant, and sin(5π/6) = sin(π/6).
Step 6: The angle that corresponds to sin(5π/6) in the range of arcsin is π/6.
Step 7: Therefore, sin^(-1)(sin(5π/6)) = π/6.
Inverse Trigonometric Functions – Understanding how inverse functions, such as sin^(-1), operate within their defined ranges.
Range of the Sine Function – Recognizing that the sine function outputs values between -1 and 1, and how this affects the input to the inverse function.
Principal Value of Inverse Functions – Identifying the principal value of the inverse sine function, which is restricted to the range [-π/2, π/2].
